Finite Group Databasebeta

← C3×#24343 #10642950

Subgroups — C3×#24343 #10642950

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic
2C2×C3×#475C211yesmaximal, characteristic, complemented
3C2×C3×#440C211yesmaximal, characteristic
4C3×#2581C211yesmaximal, characteristic, complemented
5C3×#2059C212yesmaximal, complemented
6C3×#2026C212yesmaximal, complemented
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1985C211yesmaximal, characteristic, complemented
9C3×#1877C212yesmaximal
10C3×#1864C212yesmaximal
11C3×#1804C211yesmaximal, characteristic
12#24343C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C3×#554—24no—
15C3×#554—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C3×#475—24no—
26C3×#440—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#97C2211yescharacteristic
35C3×#400—21no—
36C3×#395—21no—
37C3×#391C2212yes—
38C3×#389C2212yes—
39C3×#387—22no—
40C3×#382C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#554C2×C311yescharacteristic
47C2×#475C2×C311yescharacteristic, complemented
48C2×#440C2×C311yescharacteristic
49#2581C2×C311yescharacteristic, complemented
50#2059C2×C312yescomplemented
51#2026C2×C312yescomplemented
52#1993C2×C311yescharacteristic, complemented
53#1985C2×C311yescharacteristic, complemented
54#1877C2×C312yes—
55#1864C2×C312yes—
56#1804C2×C311yescharacteristic
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C2×C3×SD16Q814yescomplemented
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2312yes—
68C22×C3×C8C2311yescharacteristic
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yes—
71C3×#125—24no—
72C3×#125—24no—
73C3×C22⋊Q8—28no—
74C3×C22⋊Q8—216no—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C3×C4×D4—22no—
82C3×C4×D4D414yes—
83C2×C3×C4⋊C4—28no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4D412yes—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2312yes—
99C2×C3×C42C2311yescharacteristic
100C3×#107—22no—
101C3×#106—24no—
102C3×C4⋊C8—22no—
103C3×#103—22no—
104C3×#103—24no—
105C3×#102—22no—
106C3×#102—24no—
107C3×#97—22no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—22no—
111C3×#95—22no—
112C3×#95—22no—
113C22×SD16C22×C311yescharacteristic
114#554—24no—
115#554—24no—
116C2×C4⋊Q8C22×C311yescharacteristic
117C2×#125C22×C312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—22no—
120C2×C4×Q8C22×C311yescharacteristic
121C2×C4×Q8C22×C311yescharacteristic
122C2×C4×D4C22×C312yes—
123C2×C4×D4C22×C311yescharacteristic
124C22×C4⋊C4—21no—
125#475—24no—
126#440—24no—
127C2×#107C22×C311yescharacteristic
128C2×#106C22×C312yes—
129C2×C4⋊C8C22×C311yescharacteristic
130C2×#103C22×C312yes—
131C2×#103C22×C311yescharacteristic
132C2×#102C22×C312yes—
133C2×#102C22×C311yescharacteristic
134C2×#97C22×C311yescharacteristic
135#400—21no—
136#395—21no—
137#391C22×C312yes—
138#389C22×C312yes—
139#387—22no—
140#382C22×C311yescharacteristic
141#340C22×C312yes—
142#340C22×C312yes—
143#340C22×C312yes—
144#336C22×C312yes—
145#336C22×C312yes—
146C24×C3—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—42no—
155C2×C3×Q8—82no—
156C2×C3×Q8C2×Q812yes—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4C2×D412yes—
161C2×C3×D4C2×Q812yes—
162C2×C3×D4#4112yes—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—28no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—44no—
171C22×C3×C4—84no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C3×SD16—28no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—22no—
186C2×C3×C8C2×Q812yes—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—44no—
199C3×C4⋊C4—42no—
200C3×C4⋊C4C2×D414yes—
201C3×C4⋊C4C2×D412yes—
202C3×C22⋊C4—28no—
203C3×C22⋊C4—24no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—42no—
207C3×C42C2×D412yes—
208C3×C42C2×D412yes—
209C22×Q8—21no—
210C22×Q8C3×D411yescharacteristic
211C22×Q8C23×C311yescharacteristic
212C22×D4C23×C311yescharacteristic
213C23×C4—21no—
214C23×C4—22no—
215C2×SD16—22no—
216C2×SD16—22no—
217C2×SD16C3×Q814yescomplemented
218C22×C8C23×C311yescharacteristic
219C22×C8C23×C312yes—
220C22×C8C23×C311yescharacteristic
221C4⋊Q8—22no—
222C4⋊Q8C3×D414yes—
223#125—24no—
224#125—24no—
225C22⋊Q8—28no—
226C22⋊Q8—216no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—22no—
234C4×D4C3×D414yes—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4C3×D411yescharacteristic
242C2×C4⋊C4C3×D412yes—
243C2×C4⋊C4C23×C311yescharacteristic
244C2×C4⋊C4C23×C312yes—
245C2×C4⋊C4C23×C311yescharacteristic
246C2×C4⋊C4C23×C312yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C42—21no—
250C2×C42C23×C312yes—
251C2×C42C23×C311yescharacteristic
252#107—22no—
253#106—24no—
254C4⋊C8—22no—
255#103—22no—
256#103—24no—
257#102—22no—
258#102—24no—
259#97—22no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—22no—
264#95—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3C22×D411yescharacteristic
271C3×Q8—22no—
272C3×Q8—22no—
273C3×Q8—44no—
274C3×Q8—41no—
275C3×Q8—41no—
276C3×Q8—84no—
277C3×D4—22no—
278C3×D4—22no—
279C3×D4—22no—
280C3×D4C22⋊Q814yes—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—48no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—44no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—88no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12012yes—
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×Q811yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C42−1+411yescharacteristic
309C2×C3×C42−1+411yescharacteristic
310C3×C8—22no—
311C3×C8—41no—
312C3×C8—42no—
313C24—21no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—42no—
322C2×Q8—82no—
323C2×Q8C2×C3×Q812yes—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4C2×C3×D412yes—
328C2×D4C2×C3×Q812yes—
329C2×D4C3×#4112yes—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—44no—
338C22×C4—84no—
339C22×C4C2×C3×D412yes—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C24×C311yescharacteristic
344SD16—28no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—22no—
353C2×C8C2×C3×Q812yes—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—44no—
366C4⋊C4—42no—
367C4⋊C4C2×C3×D414yes—
368C4⋊C4C2×C3×D412yes—
369C22⋊C4—28no—
370C22⋊C4—24no—
371C42—22no—
372C42—22no—
373C42—42no—
374C42C2×C3×D412yes—
375C42C2×C3×D412yes—
376C22×C3—24no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3C2×#12011yescharacteristic
382C22×C3#54811yescharacteristic
383C22×C3D4×Q811yescharacteristic
384C22×C3#57512yes—
385C22×C3#57712yes—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—42no—
389C3×C4—42no—
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—44no—
393C3×C4—82no—
394C3×C4—81no—
395C3×C4C2×C22⋊Q811yescharacteristic
396C3×C4#53611yescharacteristic
397C3×C4#54811yescharacteristic
398C3×C4D4×Q811yescharacteristic
399C23—22no—
400C23—22no—
401C23—22no—
402C23—42no—
403C23—42no—
404C23C22×C3×D411yescharacteristic
405Q8—22no—
406Q8—22no—
407Q8—44no—
408Q8—41no—
409Q8—41no—
410Q8—84no—
411D4—22no—
412D4—22no—
413D4—22no—
414D4C3×C22⋊Q814yes—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—48no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—88no—
436C2×C4C3×#12012yes—
437C2×C4C3×#12012yes—
438C2×C4C22×C3×D411yescharacteristic
439C2×C4C22×C3×D411yescharacteristic
440C2×C4C22×C3×Q811yescharacteristic
441C2×C4C2×C3×#4111yescharacteristic
442C2×C4C3×2−1+411yescharacteristic
443C2×C4C3×2−1+411yescharacteristic
444C8—22no—
445C8—41no—
446C8—42no—
447C2×C3—44no—
448C2×C3#240911yescharacteristic
449C2×C3#300511yescharacteristic
450C2×C3#300911yescharacteristic
451C2×C3#337612yes—
452C2×C3#338812yes—
453C22—24no—
454C22—44no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22C2×C3×#12011yescharacteristic
459C22C3×#54811yescharacteristic
460C22C3×D4×Q811yescharacteristic
461C22C3×#57512yes—
462C22C3×#57712yes—
463C4—22no—
464C4—22no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—44no—
470C4—81no—
471C4—82no—
472C4C2×C3×C22⋊Q811yescharacteristic
473C4C3×#53611yescharacteristic
474C4C3×#54811yescharacteristic
475C4C3×D4×Q811yescharacteristic
476C3#2434311yescharacteristic, direct product factor, Sylow
477C2—44no—
478C2C3×#240911yescharacteristic
479C2C3×#300511yescharacteristic
480C2C3×#300911yescharacteristic
481C2C3×#337612yes—
482C2C3×#338812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#475C2semidirect
4. C3×#2581C2semidirect
5. C3×#2059C2semidirect
6. C3×#2026C2semidirect
7. C3×#1993C2semidirect
8. C3×#1985C2semidirect
12. #24343C3direct (N × H)
47. C2×#475C2×C3semidirect
49. #2581C2×C3semidirect
50. #2059C2×C3semidirect
51. #2026C2×C3semidirect
52. #1993C2×C3semidirect
53. #1985C2×C3semidirect
65. C2×C3×SD16Q8semidirect
217. C2×SD16C3×Q8semidirect
476. C3#24343direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#5542
C2×C3×#4752
C2×C3×#4402
C3×#25812
C3×#20592
C3×#20262
C3×#19932
C3×#19852
C3×#18772
C3×#18642
C3×#18042
#243433